NDLI: The Classical Aristotelian Hexagon Versus the Modern Duality Hexagon

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The Classical Aristotelian Hexagon Versus the Modern Duality Hexagon

Content Provider	Springer Nature Link
Author	Smessaert, Hans
Copyright Year	2011
Abstract	Peters and Westerståhl (Quantifiers in Language and Logic, 2006), and Westerståhl (New Perspectives on the Square of Opposition, 2011) draw a crucial distinction between the “classical” Aristotelian squares of opposition and the “modern” Duality squares of opposition. The classical square involves four opposition relations, whereas the modern one only involves three of them: the two horizontal connections are fundamentally distinct in the Aristotelian case (contrariety, CR vs. subcontrariety, SCR) but express the same Duality relation of internal negation (SNEG). Furthermore, the vertical relations in the classical square are unidirectional, whereas in the modern square they are bidirectional. The present paper argues that these differences become even bigger when two more operators are added, namely the U ( $${{\equiv} {\rm A}\,{\vee} \,{\rm E} }$$ , all or no) and Y ( $${\equiv{\rm I} \,{\wedge} \,{\rm O}}$$ , some but not all) of Blanché (Structures Intellectuelles, 1969). In the resulting Aristotelian hexagon the two extra nodes are perfectly integrated, yielding two interlocking triangles of CR and SCR. In the duality hexagon by contrast, they do not enter into any relation with the original square, but constitute a independent pair of their own, since they are their own SNEGs. Hence, they not only stand in a relation of external NEG, but also in one of duality. This reflexive nature of the SNEG will be shown to result in defective monotonicity configurations for the pair, namely the absence of right-monotonicity (on the predicate argument). In the second half of the paper, we present an overview of those hexagonal structures which are both Aristotelian and Duality configurations, and those which are only Aristotelian.
Starting Page	171
Ending Page	199
Page Count	29
File Format	PDF
ISSN	16618297
Journal	Logica Universalis
Volume Number	6
Issue Number	1-2
e-ISSN	16618300
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2011-09-06
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Logical square logical hexagon aristotelian relations of opposition duality relations external versus internal negation monotonicity properties modal logic Modal logic Classical first-order logic Logic of natural languages Boolean algebras Monotone operators and generalizations Computer Science Mathematics Logic
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Logic

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